The Mathematical Statistics Test Paper will have a compulsory part (Section A) of 75% of the total weightage, comprising of Mathematics(25% weightage) and Statistics (50% weightage). There will also be an optional part consisting of Section B (Mathematics: 25% weightage) and Section C (Statistics 25% weightage). Candidates seeking admission to either of the two programmes: M.Sc. in Applied Statistics & Informatics at IIT Bombay or M.Sc. in Statistics & Informatics at IIT Kharagpur will be required to attempt only Section B (Mathematics) in the optional part, whereas candidates seeking admission to the programme M.Sc. in Statistics at IIT Kanpur will be required to attempt only Section C (Statistics) in the optional part.
All the objective type questions will be in the compulsory section A only.

SECTION A
MATHEMATICS(weightage: 25%)
Sequences & Series and Differential Calculus: Convergence of sequences of real numbers. Comparision, root and ratio tests for convergence of series of real numbers. Limit, continuity, intermediate value property and differtiability. Rolle's theorem, Lagrange’s mean value theorem. Maxima and Minima, functions of a single variable.
Integral Calculus: Fundamental theorem of Integral Calculus, Application of definite integrals.
Matrices: Algebra of matrices. Rank, determinant, inverse of a matrix. Systems of linear equations Eigenvalues and eigenvectors, Cayley-Hamilton theorem.

STATISTICS (weightage: 50%)
Probability: Axiomatic definition of probability and properties. Conditional probability, multiplication rule. theorem of total probability. Bayes’ theorem and independence of events.
Random Varibles: Distribution functions. Probability mass function and probability density function. Distribution of a function of a random variable. Mathematical expection. Moments and moment generating function. Chebyshev’s inequality
Standard Distribution: Binomial, negative binomial, geometric, Poisson, Hypergeometric, uniform, exponential, gamma, beta and normal distributions. Piosson and normal approximations of a binomial distribution.
Joint Distribution: Joint, marginal and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.
Sampling distribution: Chi-square, t and F distributions, and their properties.
Estimation and Testing: Unbiased estimators. Method of moments, method of maximum likelihood. Tests of hypotheses for binomial proportion(s), mean(s) and variance(s) of normal population(s) and associated confidence intervals.

SECTION B
MATHEMATICS(WEIGHTAGE: 25%)
Functions of two and three Variables: Limit, continuity and differentiability. Partial derivatives. Maxima and minima. Double and triple integrals. Surface areas and volumes.
Differential Equation: Ordinary differential equations of the order of the form
Linear differential equations of the second order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.
Linear Algebra and Algebra: Dimention of a vector space, linear transformations. Group, normal subgroups. Lagrange’s theorem for finite groups.

SECTION C
STATISTICS(WEIGHTAGE: 25%)
Limit Theorems: Weak law of large numbers. Central limit theorem(i.i.d. with finite variance case only). Point Estimation: Minimum mean square error estimators. Sufficient statistic. Facorization theorem. Complete statistic. Rao-Blackwell theorem. Uniformly minimum variance unbiased estimators. Cramer-Rao inequatlity.
Testing of Hypotheses: Basic concepts and simple applications of Neyman-Pearson Lemma for testing simple and composite hypotheses. Likelihood ratio test for parameters of univariate normal distributions.
Interval Estimation: Concepts of confidence intervals and confidence coefficients. Confidence intervals for the parameters of univariate normal, two-independent normal and one parameter (mean) exponential distributions.